Trigonometric functions, their properties and graphics presentation. Graphs of trigonometric functions - presentation. Inverse trigonometric functions

slide 6

Trigonometric functions of a double angle:

sin 2x = 2sinx cosx cos 2x = cos2x -sin2x tg 2x = 2tg x /(1-tg2x) ctg 2x = ctg2x-1/(2 ctg x)

Slide 7

Trigonometric half angle functions

Often useful are formulas that express the powers of sin and cos of a simple argument in terms of sin and cos of a multiple, for example: The formulas for cos2x and sin2x can be used to find the values ​​of T. f. half argument

Slide 8

Trigonometric functions of the sum of angles

sin(x+y)= sin x cos y + cos x sin y sin(xy)= sin x cos y - cos x sin y cos(x+y)= cos x cos y - sin x sin y cos(xy) = cos x cos y + sin x sin y

Slide 9

For large values ​​of the argument, one can use the so-called reduction formulas, which allow one to express the T. f. any argument through T. f. argument x, which simplifies the compilation of tables T. f. and use of them, as well as the construction of graphs. These formulas look like this: in the first three formulas, n can be any integer, with the upper sign corresponding to the value n = 2k, and the lower sign corresponding to the value n = 2k + 1; in the latter - n can only be an odd number, and the upper sign is taken at n = 4k + 1, and the lower one at n = 4k - 1.

Slide 10

The most important trigonometric formulas are the addition formulas expressing the T. f. the sum or difference of the values ​​of the argument through T. f. these values: the signs in the left and right parts of all formulas are consistent, that is, the upper (lower) sign on the left corresponds to the upper (lower) sign on the right. From them, in particular, formulas are obtained for T. f. multiple arguments, for example:

slide 11

Derivatives of all Trigonometric functions are expressed in terms of Trigonometric functions

slide 12

The graph of the function y = sinx looks like:

slide 13

The graph of the function y = cosx looks like:

Slide 14

The graph of the function y = tgx looks like:

slide 15

The graph of the function y = ctgx looks like:

slide 16

The history of the emergence of trigonometric functions

T. f. arose for the first time in connection with research in astronomy and geometry. The ratios of the segments in a triangle and a circle, which are essentially T. f., are found already in the 3rd century. BC e. in the work of mathematicians Ancient Greece- Euclid, Archimedes, Apollonius of Perga, etc. However, these ratios are not an independent object of study for them, so T. f. as such they have not been studied. T. f. were initially considered as segments and were used in this form by Aristarchus (end of the 4th - 2nd half of the 3rd centuries BC)

Slide 17

Hipparchus (2nd century BC), Menelaus (1st century AD) and Ptolemy (2nd century AD) when solving spherical triangles. Ptolemy compiled the first table of chords for acute angles at intervals of 30" with an accuracy of 10-6. The expansion of the thermodynamic function into power series was obtained by I. Newton (1669). The theory of thermofunctional functions was brought into modern form by L. Euler (18th century BC). He owns the definition of TF for real and complex arguments, the symbolism now accepted, the establishment of a connection with the exponential function, the orthogonality of the system of sines and cosines

View all slides

Trigonometric functions

x = cost. Presentation on the topic: "Trigonometric functions." Number circle. All numbers with a denominator of 4 correspond to Cartesian coordinates. Up to a sign, depending on the quarter in which the point is located. Sine, cosine, tangent and cotangent. Quarter signs: Properties of sine, cosine, tangent and cotangent. Basic trigonometric formulas. Connection between trigonometric functions of angular and numerical argument. arc length AM - numeric argument, Angle. – Angular argument. Values ​​of trigonometric functions. Training exercises. Point P divides the third quarter in a ratio of 1: 5. Find the length of the arc CP, PD, AP. - Trigonometric functions.ppt

Examples of trigonometric functions

trigonometric functions. Trigonometric functions of an acute angle. Right triangle ABC. For some angles, exact values ​​can be recorded. Connection of trigonometric functions of an acute angle. Trigonometric functions of a double angle. Trigonometric functions of a half angle. Trigonometric functions of the sum of angles. You can use the so-called reduction formulas. The most important trigonometric formulas are addition formulas. Derivatives of all trigonometric functions. Graph of the function y = sinx. Graph of the function y = cosx. Graph of the function y = tgx. - Examples of trigonometric functions.ppt

Basic trigonometric functions

trigonometric functions. Mathematical model. Definition of even and odd functions. Domain. Set of values ​​of trigonometric functions. Find the scope of the function. Function scope. The set of function values. Periodicity. Which of the functions is even. g(x) function. Meaning. positive period. Function properties. Function graph. Properties of the function y=sin x. Dots. x values. Gaps. Value area. Plot the function graph. Function properties y = tg(x). Function y=tg(x). Find the domain of definition. Define a function using a formula. - Basic trigonometric functions.ppt

Algebra "Trigonometric functions"

Handbook of Algebra and the Beginnings of Analysis. Content. Trigonometry. sine and cosine. Tangent and cotangent. Trigonometric functions of a numerical argument. Trigonometric functions of angular argument. Casting formulas. Table of values ​​of trigonometric functions of some angles. Formulas for the transformation of trigonometric functions. Formulas for converting the product of trigonometric functions into a sum. Converting sums of trigonometric functions to products. Complementary Angle Formula. Arcsine. Solution of the simplest trigonometric equations. Homogeneous trigonometric equations. - Algebra "Trigonometric functions".ppt

Properties of trigonometric functions

Properties of trigonometric functions. Math cafe. Crossword. Defining each function property. Gymnastics for the eyes. Read the graph of the function. Reading a graph of a function. Fizkultminutka. List properties. The task. - Properties of trigonometric functions.ppt

Trigonometric functions and their properties

What are the similarities and differences between trigonometric functions? Problematic issue: Educational project on the topic: You, me and trigonometry. trigonometric functions. Definition. Trigonometric functions Number circle. Number circle equation: x2 + y2 = 1. Movement along the number circle is counterclockwise. Trigonometric functions Sine and cosine. Trigonometric functions Tangent and cotangent. Trigonometric functions of a numerical argument. Trigonometric functions Function y = sin x. The line that serves as a graph of the function y \u003d sin x is called a sinusoid. - Trigonometric functions and their properties.ppt

Trigonometric functions of angular argument

Values ​​of trigonometric functions of the angular argument. Summarize and systematize the educational material on the topic. Trigonometric functions of a numerical argument. The cosine of the angle A (cos A) is the abscissa (x) of the point. Values ​​of trigonometric functions of the angles of the unit circle. Values ​​of trigonometric functions of basic angles. The values ​​of the trigonometric functions of the remaining corners of the table. Signs of trigonometric functions in quarters of the unit circle. Casting formulas. The task. Independent work. - Trigonometric functions of angular argument.ppt

Graphs of trigonometric functions

Graphs of trigonometric functions. trigonometric functions. The graph of the function y \u003d sin x is a sinusoid. y=sinx. Properties of the function y \u003d sin x. y = sinx. Function properties y=sin x. 6. Intervals of monotonicity: the function increases on intervals of the form: [-p/2+2pn; p/2+2pn], n?Z. Intervals of monotonicity: the function decreases on intervals of the form: , n?Z. Properties of the function y \u003d sin x. 7. Extreme points: Хmax= p/2 +2pn, n?Z Хmin= -p/2+2pn, n?Z. 8. Range of values: E(y) = [-1;1]. Converting graphs of trigonometric functions. Plot the Function y=sin(x+p/4). - Graphs of trigonometric functions.ppt

Converting Trigonometric Plots

Converting graphs of trigonometric functions. Characterization of transformations of graphs of functions. Stretching. Function graph. Compression. Graph of the function y=f(x). Parallel transfer. Graph of the function y=f(x)+m. Transfer. Y=f(x). Graph of the function y=f(|x|). Part of the chart. Graph of the function y=|f(x)|. Sections of the resulting graph. Graph of the function y=|f(|x|)|. Characteristic of the graph of harmonic oscillation. sine function. cosine function. Tangent function. cotangent function. - Convert trigonometric graphs.ppt

Plotting Trigonometric Functions

Chart conversion. Formation of knowledge. Application of MS Excel program. Function graphs. Plotting a function graph. Parallel transfer of the schedule. Building a graph. Moving the graph along the x-axis. Y2 = sinx + 2. Y1 = sinx. Y = sin(x + 1.5) +2. Construction. Y=af(x). Y2 = 2sinx. Y = 2sin(x + 1.5) + 2. Build your own graphs. Y \u003d sin (x - 0.75) + 2. Y \u003d 2.5 cos (x + 1.5) -1. Graph of the function y=f(x + t) + m. - Plotting trigonometric functions.ppt

Transforming Graphs of Trigonometric Functions

Lesson presentation “Graphs of trigonometric functions. Graph conversion. Lesson equipment: computer, projector, screen. Objectives: To generalize knowledge and skills. Develop the ability to observe, compare, generalize. Cultivate cognitive activity, perseverance in achieving the goal. Introduction by the teacher. Let us dwell in detail on the graphs of trigonometric functions. "Graphs of Trigonometric Functions". Overview of trigonometric functions. Y=sinx Y=cosx. The first student. 1.Sine function. 2. The cosine function. Second student. Overview of trigonometric functions. y=tgx y=ctgx. - Convert graphs of trigonometric functions.ppt

y sinx function

Properties and graph of the SINUS function. Oral workout. cos90°. sin90°. sin(?/4). cos180°. sin270°. sin(?/3). cos(?/6). cos360°. ctg(?/6). tg(?/4). sin(3?/2). cos(2?). cos(-?/2). cos(?/3). cos(??). Name the functions whose graphs are shown in the figure. y = cox. Plotting y = sin x. y==sinx. P - six cells. Plotting the function y = sinx using a trigonometric circle. P - three cells. Creation of a graph template for the function y = sinx. Sine axis. sin0 \u003d 0. sinp \u003d 0. sin (-p) \u003d 0. The main properties of the function y \u003d sinx. Domain. - The set R of all real numbers. - Function y sinx.pptx

Function y=cos x

Function y = cosx. Plotting the function y = cos x. Building a graph. How to use periodicity and parity in plotting. Let's find several points for plotting. We extend the resulting graph on the entire number line. Function graph. How to find the domain of definition. Domain. Lots of meanings. Periodicity. Even, odd. Increasing, decreasing. Function zeros, positive and negative values. Properties of the function y = cos x. Transformation of the graph of the function y = cos x. Y = cos x + A. Y = cos x + A (properties). Y = k cos x. Y = k cos x (properties). - Function y=cos x.ppt

Tangent function

Properties of the function y \u003d tg x and its graph. Lesson goals. Region definitions. The function y=tg x is increasing. Plotting the function y=tg x. Function properties y=tg x. The function y=tgx is not defined. The set of function values. Find all roots of the equation. Find all solutions of the inequality. - Tangent function.ppt

Tangent and cotangent functions

Function properties. Function y = tgx. Schedule. Fraction. Building a graph. Basic properties. Meaning. Equation roots. Solutions. Numbers. Function properties y=tgx. y=ctgx. Basic properties of the function. Graph of the function y=ctgx. - Tangent and cotangent functions.ppt

Arcfunctions

Reverse trigonometric functions. Function. Equality. trigonometric functions. Domain. Function scope. Definition. Arccos t. Arctg t. Arcctg t = a. Definitions. Value area. The set of real numbers. Y \u003d arcctgx. Arccosx. Expression. Find the meaning of expressions. Arctgx. Properties of arc functions. Graphical method for solving equations. Functional-graphical method for solving equations. - Arcfunctions.ppt

Inverse trigonometric functions

Inverse trigonometric functions. From the history of trigonometric functions. Ancient Greece.III century BC. e. Euclid, Apollonius of Perga. The ratio of the sides in a right triangle. OK. 190 BC e Hipparchus of Nicaea. Abu al-Waf introduced the trigonometric functions tangent and cotangent. Karl Scherfer introduced modern notation for inverse trigonometric functions. The function y = arcsinx is strictly increasing. Function properties y = arcsin x. The arccosine of the number m is such an angle x for which: The function y= arccosx is strictly decreasing. Function properties y = arccos x . - Inverse trigonometric functions.ppt

Properties of inverse trigonometric functions

Elective course in mathematics. Inverse trigonometric functions. Solution of equations. Research. Calculate. oral exercises. Specify the scope of the function. Specify the range of the function. Find the value of the expression. Solution. Let's solve the system of equations. Term. The original equation. The triple satisfies the original equation. Repetition. Arcfunctions. Group work. Solve equations. -

Contents 1. Introduction slide 2. Beginning of the study slide 3. Stages of study slide 4. Function groups slide 5. Definition and plot of sine slide 6. Definition and plot of cosine slide 7. Definition and plot of tangent slide 8. Definition and plot of cotangent slide 9. Inverse three functions slide 10. Basic formulas slide 11. The value of trigonometry slide 12. Literature used slide 13. Author and compiler of the slide

In ancient times, trigonometry arose in connection with the needs of astronomy, surveying and construction, that is, it was purely geometric in nature and represented mainly the “calculus of chords”. Over time, some analytical points began to intersperse into it. In the first half of the 18th century there was a sharp turn, after which trigonometry took a new direction and shifted towards mathematical analysis. It was at this time that trigonometric dependencies began to be considered as functions. This has not only mathematical and historical, but also methodological and pedagogical interest. In ancient times, trigonometry arose in connection with the needs of astronomy, surveying and construction, that is, it was purely geometric in nature and represented mainly the “calculus of chords”. Over time, some analytical points began to intersperse into it. In the first half of the 18th century there was a sharp turn, after which trigonometry took a new direction and shifted towards mathematical analysis. It was at this time that trigonometric dependencies began to be considered as functions. This has not only mathematical and historical, but also methodological and pedagogical interest.

At present, much attention is paid to the study of trigonometric functions precisely as functions of a numerical argument in the school course of algebra and the beginnings of analysis. There are several different approaches to teaching this topic in a school course, and a teacher, especially a beginner, can easily become confused as to which approach is the most appropriate. But trigonometric functions are the most convenient and visual means for studying all the properties of functions (before applying the derivative), and in particular such a property of many natural processes as periodicity. Therefore, their study should be given close attention.

In addition, great difficulties in studying the topic "Trigonometric functions" in the school course arise due to the discrepancy between the sufficiently large amount of content and the relatively small number of hours allocated for the study of this topic. Thus the problem of this research work consists in the need to eliminate this discrepancy through careful selection of content and development effective methods presentation of this material. The object of the research is the process of studying the functional line in the high school course. The subject of the study is the methodology for studying trigonometric functions in the course of algebra and the beginning of analysis in the classroom.

Trigonometric functions Trigonometric functions are mathematical functions of an angle. They are important in the study of geometry, as well as in the study of periodic processes. Trigonometric functions are usually defined as the ratio of the sides of a right triangle or the lengths of certain segments in the unit circle. More modern definitions express trigonometric functions in terms of sums of series or as solutions of some differential equations, which makes it possible to expand the domain of definition of these functions to arbitrary real numbers and even to complex numbers.

In the study of trigonometric functions, the following stages can be distinguished: I. The first acquaintance with the trigonometric functions of the angular argument in geometry. The value of the argument is considered in the range (0o;90o). At this stage, students learn that the sin, cos, tg and ctg of an angle depend on its degree measure, get acquainted with tabular values, the basic trigonometric identity and some reduction formulas. II. Generalization of the concepts of sine, cosine, tangent and cotangent for angles (0o; 180o). At this stage, the relationship between trigonometric functions and the coordinates of a point on the plane is considered, the theorems of sines and cosines are proved, and the issue of solving triangles using trigonometric relations is considered. III. Introduction of the concepts of trigonometric functions of a numerical argument. IV. Systematization and expansion of knowledge about the trigonometric functions of a number, consideration of function graphs, research, including with the help of a derivative.

There are several ways to define trigonometric functions. They can be divided into two groups: analytic and geometric. 1. Analytical methods include the definition of the function y \u003d sin x as a solution to the differential equation f (x) \u003d -c * f (x) or as the sum of a power series sin x \u003d x - x3 / 3! + x5 / 5! - ... 2. Geometric methods include the definition of trigonometric functions based on projections and coordinates of the radius vector, the definition through the ratio of the sides of a right-angled triangle and the definition using a numerical circle. In the school course, preference is given to geometric methods due to their simplicity and clarity.

slide 1

slide 2

slide 3

slide 4

slide 5

Slide 7

Slide 8

Slide 9

slide 10

slide 11

slide 12

slide 13

slide 14

X y 1 y= cosx Individual survey (review of the previous day's materials)

On the site I found an interesting material “Model of biorhythms”. To build a model of biorhythms, you must enter the date of birth of a person, the date of reference (day, month, year) and the duration of the forecast (number of days). As you can see, the graph is a sinusoid.

On the site I found material that the trajectory of a bullet coincides with a sinusoid. The figure shows that the projections of the vectors on the X and Y axes, respectively, are equal to υ x = υ o cos α υ y = υ o sin α

On the site math.ru/load/shkolnaja_matematika/alge bra_10_klass/grafiki_trigon/ there is material about the 360° rotation of the Earth in 365 days. Interestingly, this can be represented as a sinusoid. math.ru/load/shkolnaja_matematika/algebra_10_klass/grafiki_trigon/

In physics lessons, we studied the oscillatory motion of a pendulum. On the site I found material that the pendulum swings along a curve called cosine

Anatole France Learning can only be fun... To digest knowledge, you must absorb it with appetite. Dinner.

Properties of the function 1. D(tg x) = R, except for x = P/2 + Pn, 2. E (tg x) = R. 3. Periodic function with the main period T = P. 4. Odd function. 5. Increases over the entire domain of definition 6. Zeros of the function: y(x) = 0 at x= Pn, 7. Not limited either from above or from below. 8. There is no maximum or minimum value. Graph of the function y=tg x.

Properties of the function y \u003d сtg x 1. D (сtg x) \u003d R, except for x \u003d Pn, 2. E (сtg x) \u003d R. 3. Periodic function with the main period T \u003d P. 4. Odd function. 5. Decreases over the entire domain of definition 6. Zeros of the function: y(x) = 0 for x = P/2 + Pn, 7. Not limited either from above or from below. 8. There is no maximum or minimum value.

It might be useful to read:

• From the revolutionary market to the realm of counterfeit: which one will be remembered for "Gorbushka";
• Usmanov awarded the winners of the meme contest about the dispute with Navalny Usmanov presented the iPhone to four winners of the meme contest;
• Thrush bird: description with photo and video, listen to bird singing, black and white (Siberian) thrush Show bird thrush;
• In Sweden, a criminal case was opened against the "daughter" of Bombardier Promised, but did not sell;
• Vargas Properties Inc.;
• Why roosters fight How to make roosters not fight;
• The behavior of planktonic organisms What does it consist of;
• Dodo or dodo bird: description and interesting facts Report on the reasons for the extinction of dodo bird species;